If the three points $(1,a,b),$ $(a,2,b),$ $(a,b,3)$ are collinear, what is the value of $a + b$?
Note that the $z$-coordinate of both $(1,a,b)$ and $(a,2,b)$ is $b,$ so the whole line must lie in the plane $z = b.$  Hence, $b = 3.$

Similarly, the $x$-coordinate of both $(a,2,b)$ and $(a,b,3)$ is $a,$ so the whole line must lie in the plane $x = a.$  Hence, $a = 1,$ so $a + b = \boxed{4}.$